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   -> Volume 2, Issue 10


Phd Thesis available
 
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Cormac Herley (cormac@gothic.research.att.com)
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PostPosted: Fri Nov 29, 2002 3:32 pm    
Subject: Phd Thesis available
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Phd Thesis available

Author: Cormac Herley
Advisor: Martin Vetterli
Title: Wavelets and Filter Banks
Institution: Columbia University

Copies can be obtained via anonymous ftp from: ftp.ctr.columbia.edu
directory: CTR-Research/advent/public/papers/PhD-theses/Herley

Abstract:

Recent work has made it clear that the design of multirate filter
banks for Signal Processing, and of wavelet bases for the analysis of
functions address essentially two versions of the same problem: construction
of structured bases for the linear expansion of signals. In the filter bank
case the signals are elements of some sequence space, while in the wavelet
case they are from some function space, but the objectives, and designs
in both cases are very similar.

The work of this thesis is to firmly establish the relation between these two
fields, to use filter bank designs to derive novel bases of wavelets, and to
use the structure used by the wavelet approach to better understand
filter banks.

Among the main results is a complete study of finite impulse response
filter banks, and their relation to wavelets of bounded support. Examples
of new symmetric bases of bounded support wavelets are given.

A complete constructive characterization of orthogonal infinite
impulse response filter banks is given which leads to orthonormal
wavelet bases with exponential decay. Particular cases include orthonormal
wavelets bases generated by halfband Butterworth filters, orthogonal
symmetric filter banks and wavelets, and orthonormal bases of wavelets
for the spline function spaces based on realizable filter banks.

On the open question of changing between orthogonal filter banks we show
that the transition between finite impulse response filter banks
can be achieved using a Gram-Schmidt procedure, and that the disruption
affects only the region of the transition. This also gives a complete
constructive way of finding all possible orthogonal solutions of the
finite duration filter bank problem.

A number of novel applications of filter banks are given. It is shown that
multirate structures can be used to implement realizable projection
operators and that they can perform exact interpolation of certain
types of continuous-time functions. It is demonstrated that time-varying
filter banks can be used to generate orthogonal bases where the
energy localization of the basis functions are distributed in the
time-frequency plane in an essentially arbitrary fashion, which is
useful for signal compression applications.
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